Applications of trigonometry

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Question

AB is a vertical pole and C is its middle point. The end A is on level ground and P is any point on the level ground other than A. The portion CB subtends an angle $β$ at P. If AP: AB = 2 : 1, and $\tan β = \frac{2}{k}, t h e n k =$

Moderate

Solution

$\begin{aligned} Let A C = B C = h \\ \Rightarrow A P = 4 h \end{aligned}$
$\begin{aligned} \tan α = \frac{1}{4} and \tan (α + β) = \frac{1}{2} \\ \Rightarrow \frac{\frac{1}{4} + \tan β}{1 - \frac{1}{4} \tan β} = \frac{1}{2} \Rightarrow \tan β = 2 / 9 \end{aligned}$

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